To apply Trotterization, we need to express the Hamiltonian as a sum of elementary terms, each of which can be directly exponentiated on a quantum computer; see [sec:prelimpf]Section sec:prelimpf for a review of this algorithm. For the electronic Hamiltonian (REF ), it suffices to consider the two-term decomposition \(H=T+V\) , as the exponentials of \(T\)  and \(V\)  can be directly implemented using various quantum circuits. For instance, all the terms in \(V\)  commute with each other so \(e^{-i t V}= \prod _{l,m} e^{-i t \nu _{l,m}N_l N_m }\) , where each \(e^{-i t \nu _{l,m}N_l N_m }\)  corresponds to a two-qubit operation under the Jordan-Wigner transformation. On the other hand, the exponential \(e^{-i t T}\)  can be implemented by diagonalization, i.e., \(e^{-i t T}= U e^{-i \sum \lambda _\ell N_\ell } U^\dagger \) , where \(U\)  can be efficiently implemented using Givens rotations [1]}, [2]}. In cases where \(\tau \)  and \(\nu \)  are translationally invariant \(\tau _{j,k}=\tau _{j+q,k+q}\) , \(\nu _{l,m}=\nu _{l+q,m+q}\) , we can implement \(e^{-itT}\)  using the fast fermionic Fourier transform [3]} and a related circuit implementation exists for \(e^{-itV}\)  [4]}.
